# Absolute Value Equations

Equations with a variable or variables within
absolute value
bars are known as
**
absolute value equations
**
.

When solving equations that involve absolute values, there are two cases to consider.

**
Case 1:
**
The expression inside the absolute value bars is positive.

**
Case 2:
**
The expression inside the absolute value bars is negative.

For example, consider the expression $\left|\text{\hspace{0.17em}}4x+2\text{\hspace{0.17em}}\right|=8$ .

For this to be true, either

$4x+2=8$

OR

$4x+2=-8$ .

You need to solve both equations. In this case, the solution to the first one is
$x=4$
, and the the solution to the second one is
$x=-5$
. So there are
**
two solutions
**
,
$x=4$
and
$x=-5$
.

It's also possible for an absolute value equation to have
**
one solution
**
:

$\left|\text{\hspace{0.17em}}x+3\text{\hspace{0.17em}}\right|=0$ has the single solution $x=-3$

or
**
no solutions:
**

$\left|\text{\hspace{0.17em}}5x+1\text{\hspace{0.17em}}\right|=-6$

(The absolute value of any expression is positive, so there is no value of $x$ for which this is true.)